This collaborative art piece, Yes/No, combines the mathematics of
computational origami with ideas from book art, combining words with an image
in the form of a folded paper structure. The piece consists of hand-inked
paper that looks unrecognizable when unfolded, but when folded forms the
three-dimensional form of one word while the print simultaneously comes
together to form the word's antonym. In short, if you look down on the folded
word “yes”, you see the image of “no”, and vice versa.
The completed folded structure simply negates itself by embodying an
intentionally mixed message.

Origins

We met Sarah Stengle in the summer of 2009 in Banff at BRIDGES, a conference
about the connections between mathematics, music, art, architecture, and
culture. Erik gave a talk about how we enjoy working on projects across both
mathematics and art, with new mathematical theorems and unsolved problems
inspiring new artwork, and the challenge of creating art inspiring new
mathematical questions. Central to this spirit is collaboration, both the
extensive father-son collaboration, as well as collaboration with hundreds of
other mathematicians and artists.

Sarah was intrigued by this concept of collaboration. Given her expertise in
book art, and our expertise in approaching origami mathematically, we decided
to collaborate on a book project that involved folding. Over the next several
months, we discussed via email various ideas centered on making a textual
object as a subset of book art. A desired quality was to produce something new
and spontaneous, and several ideas evolved through this brainstorming. We
discussed flipbooks, book furniture, construction toys, blown glass (we are
both glassblowers, Sarah casts glass), and new fonts. In the end we went back
to our original thought of incorporating origami, as it seemed the most
natural for us. What followed were several conversations in which we raised
the possibility of folding words out of text. Sarah suggested that it would be
wonderful to fold words that said one thing, and were printed with another
message entirely. In short, an object would convey an emotional state of being
conflicted, while being grounded in mathematics.

One of our first ideas was the concept of folding one word into another,
representing the transformation and uncertainty of ideas. For example, the
text “yes” could be printed on the paper and fold into a printed
“no”, or vice versa, representing the uncertainty of decision or
the making of a decision. We experimented with different foldings, but lacked
a general mathematical principle for designing the folds, making it difficult
to find a design with the elegance that we felt the idea required. We looked
at existing origami fonts, and agreed that it would be far more interesting to
fold entire words rather than single letters. So although we had this idea
early on, we needed the right mathematics before we could turn it into a
reality.

In summer 2010, we published a
paper, together with an MIT graduate student and origami expert Jason Ku,
about how to fold origami “mazes”—orthogonal vertical walls
protruding at equal heights from a rectangular floor—efficiently from a
rectangle of paper just a small factor larger than the floor. Also that
summer, we realized that we could use these orthogonal mazes to write letters,
and designed a font to
“write” a desired message by folding a rectangle of paper. The
result is a computer algorithm that transforms any text into a crease pattern
which folds into a 3D version of that text. In some sense, the crease pattern
encodes a secret message that can be deciphered by folding, which is often
quite a challenge. You can experiment yourself with creating such crease
patterns using an online web application, but be
warned that the folding is quite difficult.

Coming Together

The maze-folding mathematical algorithm, and the font design project, in turn
inspired us to come back to our concept with Sarah of folding one word into
another. One idea is to print a crease pattern on top of text onto a sheet of
paper, and the crease pattern folds a 3D realization of different text, and
renders the original text no longer legible. Thus the artwork could be in two
states: the simple unfolded state where the printed text is legible, and the
complex folded state where only the 3D text is legible. This piece could
represent the difficulty in making up one's mind.

To add somewhat to the mystery, we opted for a different approach: print
pieces of letters and a crease pattern onto a sheet of paper so that, by
folding, both the printed text and the 3D text become simultaneously visible.
In this way, the unfolded sheet is again a puzzle—both the printed text
and 3D text require some thought to read—but by folding, they
simultaneously come into view.

Our first piece in this series is called Yes/No, and it represents coming to
understand both sides of a decision simultaneously. We go from confusion in
seeing neither the Yes nor the No, to confusion in seeing both Yes and No, not
knowing what the answer is. Perhaps there never is one right answer. The
print is from elephant hide paper, a strong archival paper that can
withstand robotic scoring and retains it shape well when folded.

The Yes/No handout lets you try folding the crease pattern.

Handout

Accompanying the piece is an 8.5" × 11" handout
encouraging you to try folding the form.
Feel free to print it out and challenge yourself.